I don't think that's possible.
There are strict rules about who is allowed on the Senate floor, and whether those allowed are allowed to speak.
Normal Senate rules would apply. The President of the Senate would preside. If the President of the Senate is not there, the President Pro Tempore presides. If the President Pro Tempore is not available, a designated Senator presides over the day's proceedings. That's why you always see different people at the gavel on different days.
So, one way or another, without Harris presiding there must be a Democrat Senator who steps up, because no Republican would agree to do it -- not even Romney, I suspect, unless he is signaling his intent to switch parties by doing so. If he did, he would be finished as a Senator from Utah.
-PJ
Much thanks for that input PJ2. You cleared up a lot of the mystery about the process for all of us.